I would say it's a myth. Consider that turkeys are active during the day, looking for food, preening their feathers, and socializing. At night, they sleep. This is a normal cycle. There's no indication that they experience some sort of abnormal tiredness. They are just going about their business like other poultry or birds in general. So, it's not a real phenomenon.
It's a myth. There's no scientific basis to prove that turkeys are particularly 'tired' in any unique way. They follow the normal rhythms of activity and rest like most animals.